Folded Normal Distribution¶

Definition¶

Let $Y$ be a normally distributed random variable. The random variable $X=|Y|$ has a folded normal distribution.

Interpretation¶

A folded normal distribution is obtained by reflecting negative normal values to the positive side. It is useful when only the magnitude of a normally distributed quantity matters.

A normal density centred at 0.7, its mirror image, and their sum on the positive axis, which is the density of the absolute value.

The illustrated calculation gives $X=|Y|$ $∣ Y ∣$ . The sign is lost, so the density of $Y$ on the negative side is folded onto the positive side and all mass lands on $[0,\infty)$ .

Move $\mu$ towards $0$ : more of the normal density of $Y$ lies below zero and gets reflected, and the density of $X=|Y|$ is the sum of the two dashed curves.

The normal distribution supplies the original variable $Y$ . Folding by absolute value combines the probability near a positive value with the probability near its negative, giving a nonnegative transformed variable.

Key Results¶

A folded normal variable is the absolute value of a normal variable.
Its support is nonnegative.
When the underlying normal mean is zero, it is called a half-normal distribution: $X=|Y|$ then has density $2f_Y(x)$ for $x\ge0$ $0$ .

Examples¶

The absolute measurement error from a normally distributed error has a folded normal distribution.
The distance from zero of a one-dimensional normal displacement is folded normal.
Quality-control deviations recorded only by magnitude can follow a folded normal model.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 5

Aliases: folded Gaussian distribution Keywords: distribution of the absolute value of a normal random variable Symbols: infinity, mu

Last updated: 20 October 2026 03:02